Average Error: 10.0 → 0.1
Time: 58.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{1 + x}}{x} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{1 + x}}{x} \cdot \frac{2}{x - 1}
double f(double x) {
        double r6491828 = 1.0;
        double r6491829 = x;
        double r6491830 = r6491829 + r6491828;
        double r6491831 = r6491828 / r6491830;
        double r6491832 = 2.0;
        double r6491833 = r6491832 / r6491829;
        double r6491834 = r6491831 - r6491833;
        double r6491835 = r6491829 - r6491828;
        double r6491836 = r6491828 / r6491835;
        double r6491837 = r6491834 + r6491836;
        return r6491837;
}
↓
double f(double x) {
        double r6491838 = 1.0;
        double r6491839 = 1.0;
        double r6491840 = x;
        double r6491841 = r6491839 + r6491840;
        double r6491842 = r6491838 / r6491841;
        double r6491843 = r6491842 / r6491840;
        double r6491844 = 2.0;
        double r6491845 = r6491840 - r6491839;
        double r6491846 = r6491844 / r6491845;
        double r6491847 = r6491843 * r6491846;
        return r6491847;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied frac-sub26.2

    \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]

  4. Applied frac-add25.8

    \[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]

  5. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.3

    \[\leadsto \frac{\color{blue}{1 \cdot 2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

  8. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}}\]
  9. Using strategy rm

  10. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1}}{x}} \cdot \frac{2}{x - 1}\]

  11. Final simplification0.1

    \[\leadsto \frac{\frac{1}{1 + x}}{x} \cdot \frac{2}{x - 1}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))